Accurate estimate of the J integral is required in a valid experimental evaluation of J-based fracture resistance curves for ductile materials. The fracture toughness test standard ASTM E1820 allows a basic method and a resistance curve method to be used experimentally to evaluate the J values via standard specimens. The basic method obtains J estimates using the η factor method that was developed for a stationary crack. The resistance curve method obtains crack growth corrected J estimates using an incremental equation that was proposed by Ernst et al. (“Estimation on J-lntegral and Tearing Modulus T from a Single Specimen Test Record,” Fracture Mechanics: Thirteenth Conference, ASTM STP 743, 1981, pp. 476–502) for a growing crack and has been accepted as the most accurate equation available for about three decades. Recently, Neimitz (“The Jump-Like Crack Growth Model, the Estimation of Fracture Energy and JR Curve,” Eng. Fract. Mech., Vol. 75, 2008, pp. 236–252), and Kroon et al. (“A Probabilistic Model for Cleavage Fracture with a Length Scale-Parameter Estimation and Predictions of Growing Crack Experiments,” Eng. Fract. Mech., Vol. 75, 2008, pp. 2398–2417) presented two different approximate equations for the J-integral, which they proposed as more accurate than the Ernst equation. Therefore, further investigation is needed to determine a truly accurate approximation for the J-integral equation. With this objective, the present paper proposes different mathematical and physical models to approximate the J-integral equation. The physical models are developed in terms of the deformation theory and the jump-like crack growth assumption. Relations between the proposed models and the existing equations are identified. Systematic evaluations of the proposed models are then made using a theoretical procedure of J-R curves for both low and high strain hardening materials, and using experimental data from an actual single edge-notched bend specimen made of HY80 steel. Accuracy of the proposed models is determined, and a more accurate approximation of J-integral equation is thus suggested for J-R curve testing.